Problem Statement
Let $\delta(m,\alpha)$ denote the density of the set of integers which are divisible by some $d\equiv 1\pmod{m}$ with $1<d<\exp(m^\alpha)$. Does there exist some $\beta\in (1,\infty)$ such that\[\lim_{m\to \infty}\delta(m,\alpha)\]is $0$ if $\alpha<\beta$ and $1$ if $\alpha>\beta$?
Categories:
Number Theory Divisors
Progress
It is trivial that\[\delta(m,\alpha)<\frac{m^\alpha+1}{m}\to 0\]if $\alpha <1$, and Erdős claims in [Er79e] he could prove that the same is true for $\alpha=1$.This was proved in the affirmative with $\beta=1/\log 2$ by Hall [Ha92].
See also [696].
Source: erdosproblems.com/697 | Last verified: January 16, 2026